A geometric obstruction to the contact type property
Following the concept of contact type hypersurface in a symplectic manifold introduced by A. Weinstein, the author finds the geometric obstruction for a hypersurface to be of contact type. A hypersurface \(S\subset (M,\omega)\) is said to be of restricted contact type if there is a 1-form \(\lambda\) on \(S\) such that \(d\lambda= \omega|_S\), \(\lambda\wedge (d\lambda)^{n-1}\) is a volume form on \(S\), \(\dim S= 2n-1\), and \(\lambda\) extends to \(\overline{\lambda}\) with the property \(d\overline{\lambda}= \omega\). For a connected compact hypersurface \(S\) in the standard symplectic space \((\mathbb{R}^{2n}, \omega_{2n})\), \(n\geq 2\), the author proves that there exist numbers \(a,b\), \(0<a\leq b\), and an embedding \(f: [-4b,4b]\times S^1\times B^{2n-2}(4b)\to \mathbb{R}^{2n}\), such that \(f([-4b,-a]\times S^2\times B^{2n-2}(4b))\subset B(S)\) and \(f([a,4b]\times S^1\times B^{2n-2} (4b))\subset U(S)\). Then, in this case, \(S\) is not of restricted contact type. In the notation above, \(B(S)\) is the bounded and \(U(S)\) the unbounded component of \(\mathbb{R}^{2n}-S\), and \(B^k(r)\) is the closed ball around zero in \(\mathbb{R}^k\) of radius \(r\). The author obtains various corollaries of this result and shows the stronger result about the nontriviality of certain symplectic homology groups.
scientific article; zbMATH DE number 1197140